The most famous festival of the 1960s was not held in Woodstock.
The official name was the Woodstock Music and Art Fair, promoted with the line "Three Days of Peace and Music." It took place from August 15 to 18, 1969, on Max Yasgur's farm in Bethel, New York. According to Chronogram, that site is about 43 miles southwest of the town of Woodstock as the crow flies.
The name was chosen for the town's image. Britannica explains that Woodstock had been an artists' retreat since the turn of the century, and that Bob Dylan and other musicians lived there. That made it a hip name to put on a poster.
The organizers did try for the real thing. Britannica says both Woodstock and Wallkill denied permission for the festival. Wikipedia's article on Bethel adds that the event had first been planned for Wallkill and moved when Wallkill withdrew. The promoters kept the name anyway, and Yasgur offered his land after the towns refused. Sources give different crowd figures: Britannica says some 400,000 people came, and Wikipedia says nearly 500,000.
The town's own name is older and plainer. Chronogram says it is Old English and means "a clearing in the woods." The town historian told Chronogram he has "no doubt" that Robert Livingston named the New York town, but the reason is unclear. The name was already in use elsewhere: a Connecticut town was named after the Woodstock in Oxfordshire, England, and the article also lists Woodstocks in Vermont, Virginia and Georgia.
So a festival was named after a town, the town after an English original, and the English original after a clearing. None of the events of August 1969 happened in the clearing the name refers to.